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On the Convergence of WKB Approximations of the Damped Mathieu Equation

Dwight Nwaigwe

Abstract

Consider the differential equation ${ m\ddot{x} +γ\dot{x} -xε\cos(ωt) =0}$, $0 \leq t \leq T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m \to 0$ we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as $m \to 0$ in the sense that the difference in sup norm is bounded as function of $m$ for a given $T$. However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are $O(m^2)$ and $O(m)$ for the periodic parts.

On the Convergence of WKB Approximations of the Damped Mathieu Equation

Abstract

Consider the differential equation , . The form of the fundamental set of solutions are determined by Floquet theory. In the limit as we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as in the sense that the difference in sup norm is bounded as function of for a given . However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are and for the periodic parts.

Paper Structure

This paper contains 5 sections, 3 theorems, 35 equations, 1 table.

Table of Contents

  1. Abstract
  2. Introduction
  3. Characteristic Exponents
  4. Periodic Parts
  5. Acknowledgements

Key Result

Lemma 1

$\Delta(0)=1-\frac{m \pi \epsilon^2}{\gamma^3 \omega}+O(m^2)$

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof